Nucleation-free 3D rigidity

All known examples of generic 3D bar-and-joint frameworks where the distance between a non-edge pair is implied by the edges in the graph contain a rigid vertexinduced subgraph. In this paper we present a class of arbitrarily large graphs with no non-trivial vertex-induced rigid subgraphs, which have implied distances between pairs of vertices not joined by edges. As a consequence, we obtain (a) the first class of counter-examples to a potential combinatorial characterization of 3D generic independence and rigidity proposed by Sitharam and Zhou [5] and (b) the first example of a 3D rigidity circuit which has no non-trivial rigid induced subgraphs.